Additive Combinatorics and Lattice Polyhedra

Additive Combinatorics and Lattice Polyhedra

Gautami Bhowmik

Universit´e de Lille 1.

12 July, 2012

Gautami Bhowmik CAParis

The Context

In additive combinatorics one studies sets and sequences embedded in an ambient group and their behaviour under the group operation.

In particular, the study of zero-sum sequences arose from that of factorizations of integers in non-principal ideal domains by Davenport. The problems studied aredirector inverse:

Given a sequence of elements in a finite abelian group written additively, when does there exist a subsequence whose sum is zero ? What can be said when no such zero-sum sequence exists?

Several methods,algebraic, combinatorial, analytic.., have been used in this study.

Here we will indicate a new approach : the use ofconvex polyhedrato deal with elementarypgroups,p being a large prime.

The Context

In additive combinatorics one studies sets and sequences embedded in an ambient group and their behaviour under the group operation.

In particular, the study of zero-sum sequences arose from that of factorizations of integers in non-principal ideal domains by Davenport.

The problems studied aredirector inverse:

Given a sequence of elements in a finite abelian group written additively, when does there exist a subsequence whose sum is zero ? What can be said when no such zero-sum sequence exists?

Several methods,algebraic, combinatorial, analytic.., have been used in this study.

Here we will indicate a new approach : the use ofconvex polyhedrato deal with elementarypgroups,p being a large prime.

Gautami Bhowmik CAParis

The Context

In additive combinatorics one studies sets and sequences embedded in an ambient group and their behaviour under the group operation.

In particular, the study of zero-sum sequences arose from that of factorizations of integers in non-principal ideal domains by Davenport.

The problems studied aredirector inverse:

Given a sequence of elements in a finite abelian group written additively, when does there exist a subsequence whose sum is zero ? What can be said when no such zero-sum sequence exists?

Several methods,algebraic, combinatorial, analytic.., have been used in this study.

Here we will indicate a new approach : the use ofconvex polyhedrato deal with elementarypgroups,p being a large prime.

The Context

In additive combinatorics one studies sets and sequences embedded in an ambient group and their behaviour under the group operation.

In particular, the study of zero-sum sequences arose from that of factorizations of integers in non-principal ideal domains by Davenport.

The problems studied aredirector inverse:

Given a sequence of elements in a finite abelian group written additively, when does there exist a subsequence whose sum is zero ? What can be said when no such zero-sum sequence exists?

Several methods,algebraic, combinatorial, analytic.., have been used in this study.

Here we will indicate a new approach : the use ofconvex polyhedrato deal with elementarypgroups,p being a large prime.

Gautami Bhowmik CAParis

Examples of Applications

We now give examples where sumsets can be understood in terms of polyhedra.

Lets(Z_p^d) be the smallest integernsuch that every sequence ofn elements inZ^dp contains a zero-sum sequence of lengthp. It is known that

Erd˝os-Ginzburg-Ziv, 1962 s(Zp) = 2p−1.

and the Kemnitz conjecture gives Reiher, 2004

s(Z_p²) = 4p−3.

(Main tool : the polynomial method.) Ford≥3, we only have bounds, for example, Elsholtz, 2004 ; Alon-Dubiner, 1995

9p−8≤s(Z_p³)≤14022p.

This lower bound usesgraph theorywhile the upper isconstructive.

Examples of Applications

We now give examples where sumsets can be understood in terms of polyhedra.

Lets(Z_p^d) be the smallest integernsuch that every sequence ofn elements inZ^dp contains a zero-sum sequence of lengthp.

It is known that

Erd˝os-Ginzburg-Ziv, 1962 s(Zp) = 2p−1.

and the Kemnitz conjecture gives Reiher, 2004

s(Z_p²) = 4p−3.

(Main tool : the polynomial method.) Ford≥3, we only have bounds, for example, Elsholtz, 2004 ; Alon-Dubiner, 1995

9p−8≤s(Z_p³)≤14022p.

This lower bound usesgraph theorywhile the upper isconstructive.

Gautami Bhowmik CAParis

Examples of Applications

We now give examples where sumsets can be understood in terms of polyhedra.

Lets(Z_p^d) be the smallest integernsuch that every sequence ofn elements inZ^dp contains a zero-sum sequence of lengthp. It is known that

Erd˝os-Ginzburg-Ziv, 1962 s(Zp) = 2p−1.

and the Kemnitz conjecture gives Reiher, 2004

s(Z_p²) = 4p−3.

(Main tool : the polynomial method.)

Ford≥3, we only have bounds, for example, Elsholtz, 2004 ; Alon-Dubiner, 1995

9p−8≤s(Z_p³)≤14022p.

This lower bound usesgraph theorywhile the upper isconstructive.

Examples of Applications

We now give examples where sumsets can be understood in terms of polyhedra.

Lets(Z_p^d) be the smallest integernsuch that every sequence ofn elements inZ^dp contains a zero-sum sequence of lengthp. It is known that

Erd˝os-Ginzburg-Ziv, 1962 s(Zp) = 2p−1.

and the Kemnitz conjecture gives Reiher, 2004

s(Z_p²) = 4p−3.

(Main tool : the polynomial method.) Ford≥3, we only have bounds, for example, Elsholtz, 2004 ; Alon-Dubiner, 1995

9p−8≤s(Z_p³)≤14022p.

This lower bound usesgraph theorywhile the upper isconstructive.

Gautami Bhowmik CAParis

Our method gives (B. and Schlage-Puchta) Theorem

Let p be a large prime number. Thens(Z_p³) = (9 +o(1))p.

Concerning zero-sum free sequences, it is clear, ford= 2, that the sub-sequenceA={(0,0)^p−1,(1,0)^p−1,(0,1)^p−1,(1,1)^p−1} contains no zero-sum sequence of lengthp and this is expected to be the generic example, up to an affine base change, of a maximal zero-sum free sequence of lenthp.

Whenever a primepsatisfies this inverse property it is said to have Property D (Gao and Geroldinger, 2003). We show that

Theorem

Every large prime p has Property D.

Our method gives (B. and Schlage-Puchta) Theorem

Let p be a large prime number. Thens(Z_p³) = (9 +o(1))p.

Concerning zero-sum free sequences, it is clear, ford= 2, that the sub-sequenceA={(0,0)^p−1,(1,0)^p−1,(0,1)^p−1,(1,1)^p−1} contains no zero-sum sequence of lengthp and this is expected to be the generic example, up to an affine base change, of a maximal zero-sum free sequence of lenthp.

Whenever a primepsatisfies this inverse property it is said to have Property D (Gao and Geroldinger, 2003). We show that

Theorem

Every large prime p has Property D.

Gautami Bhowmik CAParis

Our method gives (B. and Schlage-Puchta) Theorem

Let p be a large prime number. Thens(Z_p³) = (9 +o(1))p.

Concerning zero-sum free sequences, it is clear, ford= 2, that the sub-sequenceA={(0,0)^p−1,(1,0)^p−1,(0,1)^p−1,(1,1)^p−1} contains no zero-sum sequence of lengthp and this is expected to be the generic example, up to an affine base change, of a maximal zero-sum free sequence of lenthp.

Whenever a primepsatisfies this inverse property it is said to have Property D (Gao and Geroldinger, 2003). We show that

Theorem

Every large prime p has Property D.

Our method gives (B. and Schlage-Puchta) Theorem

Let p be a large prime number. Thens(Z_p³) = (9 +o(1))p.

Concerning zero-sum free sequences, it is clear, ford= 2, that the sub-sequenceA={(0,0)^p−1,(1,0)^p−1,(0,1)^p−1,(1,1)^p−1} contains no zero-sum sequence of lengthp and this is expected to be the generic example, up to an affine base change, of a maximal zero-sum free sequence of lenthp.

Whenever a primepsatisfies this inverse property it is said to have Property D (Gao and Geroldinger, 2003). We show that

Theorem

Every large prime p has Property D.

Gautami Bhowmik CAParis

Idea of the method

We considerAa sequence ofZ^dp where pis a large prime. To prove the sort of results mentioned, we use the following steps :

1 For every ’large’ peitherAcontains the desired zero-sum or ’most’ elements in Ahave ’large’ multiplicity. (adaptation of the

Alon-Dubiner technique)

Further, the coordinates of all elements with large multiplicity are bounded. (exponential sum estimates)

2 If the statement to be proven is sensitive to an error term we verify that a potential counterexample does not contradict this. (no standard way)

3 Formulate and solve the corresponding problem on R. (geometry of numbers). and use a continuity argument to return to the original problem.

In the last step, we introduce a ’dual’ of Freiman isomorphisms. For large primes,Zp approximatesT¹ andZ^dp approximatesT^d. Sumsets become convex polyhedra and we can use geometry.

Idea of the method

We considerAa sequence ofZ^dp where pis a large prime. To prove the sort of results mentioned, we use the following steps :

1 For every ’large’ peitherAcontains the desired zero-sum or ’most’

elements in Ahave ’large’ multiplicity. (adaptation of the Alon-Dubiner technique)

Further, the coordinates of all elements with large multiplicity are bounded. (exponential sum estimates)

2 If the statement to be proven is sensitive to an error term we verify that a potential counterexample does not contradict this. (no standard way)

3 Formulate and solve the corresponding problem on R. (geometry of numbers). and use a continuity argument to return to the original problem.

In the last step, we introduce a ’dual’ of Freiman isomorphisms. For large primes,Zp approximatesT¹ andZ^dp approximatesT^d. Sumsets become convex polyhedra and we can use geometry.

Gautami Bhowmik CAParis

Idea of the method

We considerAa sequence ofZ^dp where pis a large prime. To prove the sort of results mentioned, we use the following steps :

1 For every ’large’ peitherAcontains the desired zero-sum or ’most’

elements in Ahave ’large’ multiplicity. (adaptation of the Alon-Dubiner technique)

Further, the coordinates of all elements with large multiplicity are bounded. (exponential sum estimates)

2 If the statement to be proven is sensitive to an error term we verify that a potential counterexample does not contradict this. (no standard way)

3 Formulate and solve the corresponding problem on R. (geometry of numbers). and use a continuity argument to return to the original problem.

In the last step, we introduce a ’dual’ of Freiman isomorphisms. For large primes,Zp approximatesT¹ andZ^dp approximatesT^d. Sumsets become convex polyhedra and we can use geometry.

Idea of the method

We considerAa sequence ofZ^dp where pis a large prime. To prove the sort of results mentioned, we use the following steps :

1 For every ’large’ peitherAcontains the desired zero-sum or ’most’

elements in Ahave ’large’ multiplicity. (adaptation of the Alon-Dubiner technique)

Further, the coordinates of all elements with large multiplicity are bounded. (exponential sum estimates)

2 If the statement to be proven is sensitive to an error term we verify that a potential counterexample does not contradict this. (no standard way)

3 Formulate and solve the corresponding problem on R. (geometry of numbers). and use a continuity argument to return to the original problem.

In the last step, we introduce a ’dual’ of Freiman isomorphisms. For large primes,Zp approximatesT¹ andZ^dp approximatesT^d. Sumsets become convex polyhedra and we can use geometry.

Gautami Bhowmik CAParis

Idea of the method

We considerAa sequence ofZ^dp where pis a large prime. To prove the sort of results mentioned, we use the following steps :

1 For every ’large’ peitherAcontains the desired zero-sum or ’most’

elements in Ahave ’large’ multiplicity. (adaptation of the Alon-Dubiner technique)

Further, the coordinates of all elements with large multiplicity are bounded. (exponential sum estimates)

2 If the statement to be proven is sensitive to an error term we verify that a potential counterexample does not contradict this. (no standard way)

3 Formulate and solve the corresponding problem on R. (geometry of numbers). and use a continuity argument to return to the original problem.

In the last step, we introduce a ’dual’ of Freiman isomorphisms. For large

Alon-Dubiner constant

Aprimitive polyhedron inR^d is a convex polyhedron whose vertices have integral coordinates that generateZ^d as an affine space* such that for none of itsk≤d dimensional facesF, the renormalised faceF⁰ contains any interior lattice point.

* If this is not the case, the polyhedron can be renormalised by a linear map.

Definition

LetAD_R(d)be the maximum possible number of vertices of a primitive polyhedron inR^d.

This is the real analogue of the Alon-Dubiner constant

AD(d) = lim sups(Z_p^d) p

for which it is known that AD, 1995

AD(d) is finite for all d.

Gautami Bhowmik CAParis

Alon-Dubiner constant

Aprimitive polyhedron inR^d is a convex polyhedron whose vertices have integral coordinates that generateZ^d as an affine space* such that for none of itsk≤d dimensional facesF, the renormalised faceF⁰ contains any interior lattice point.

* If this is not the case, the polyhedron can be renormalised by a linear map.

Definition

LetAD_R(d)be the maximum possible number of vertices of a primitive polyhedron inR^d.

This is the real analogue of the Alon-Dubiner constant

AD(d) = lim sups(Z_p^d) p

for which it is known that

Quantitative version

Theorem

Let d≥3be an integer, C, δ, >0 real numbers, p>p0(d,C, , δ)a sufficiently large prime.

1 If A⊆Z^dp is a multiset with |A| ≥(AD_R(d) +)p, such that all elements of A have coordinates of absolute value≤C , then A contains a zero-sum of length p.

2 If A⊆Z^dp is a multiset such that not all elements of A with the exception of ₂p have coordinates ≤C and if A does not contain a zero-sum of length p, then there exists some t <d and a multi-set B ⊆Z^tp with|B|=|A|, such that for allτ <t the intersection of B with aτ-dimensional affine subspace is of cardinality<s(Z^τ+(d−t)p ), andΣ^(1−δ)p(B)6=Z^tp.

3 We haves(Z^dp)≥(p−1)AD_R(d) + 1.

The bounds above areineffective, but could be made effective.

Gautami Bhowmik CAParis

Ford= 3, condition (2) of the last theorem becomes empty.

We also prove thatAD_R(3) = 9.

This gives the asymptotic result fors(Z_p³).

Knowing thatAD_R(2) = 4 we might expect that Conjecture

For all d , we have AD_R(d) =AD(d)

s(Z_p^d) =AD_R(d)(p−1) + 1.

Ford= 3, condition (2) of the last theorem becomes empty.

We also prove thatAD_R(3) = 9.

This gives the asymptotic result fors(Z_p³).

Knowing thatAD_R(2) = 4 we might expect that Conjecture

For all d , we have AD_R(d) =AD(d)

s(Z_p^d) =AD_R(d)(p−1) + 1.

Gautami Bhowmik CAParis

Example of a primitive polyhedron

The example considered by Harborth (1973) forp= 3, and later by Elsholtz forp≥3, was the set of lattice points



 0 0 0



,



 0 0 1



,



 0 1 0



,



 0 1 1



,



 1 0 0



,



 1 0 1



,



 1 1 2



,



 1 2 2



,



 2 1 2



.

ConsiderP, the convex hull of the above 9 points. It is in fact a primitive polyhedron.

1

Gautami Bhowmik CAParis

The only interior lattice point inP is (1,1,1), on the 2-dimensional face with vertices (0,1,0),(1,0,0),(1,2,2),(2,1,2).

The intersection of the real affine space generated by these four points withZ³ gives an affine lattice isomorphic toZ².

The vertices (0,0),(1,0),(1,2),(2,2) constitute a face containing (1,1) butdo not generateZ²(area = 2).

The renormalization when chosen to be the unit square (area = 1) no longer contains an interior lattice point.

The only interior lattice point inP is (1,1,1), on the 2-dimensional face with vertices (0,1,0),(1,0,0),(1,2,2),(2,1,2).

The intersection of the real affine space generated by these four points withZ³ gives an affine lattice isomorphic toZ².

The vertices (0,0),(1,0),(1,2),(2,2) constitute a face containing (1,1) butdo not generateZ²(area = 2).

The renormalization when chosen to be the unit square (area = 1) no longer contains an interior lattice point.

Gautami Bhowmik CAParis

The only interior lattice point inP is (1,1,1), on the 2-dimensional face with vertices (0,1,0),(1,0,0),(1,2,2),(2,1,2).

The intersection of the real affine space generated by these four points withZ³ gives an affine lattice isomorphic toZ².

The vertices (0,0),(1,0),(1,2),(2,2) constitute a face containing (1,1) butdo not generateZ²(area = 2).

The renormalization when chosen to be the unit square (area = 1) no longer contains an interior lattice point.

The only interior lattice point inP is (1,1,1), on the 2-dimensional face with vertices (0,1,0),(1,0,0),(1,2,2),(2,1,2).

The intersection of the real affine space generated by these four points withZ³ gives an affine lattice isomorphic toZ².

The vertices (0,0),(1,0),(1,2),(2,2) constitute a face containing (1,1) butdo not generateZ²(area = 2).

The renormalization when chosen to be the unit square (area = 1) no longer contains an interior lattice point.

Gautami Bhowmik CAParis